Problem 1. Stability of rigid body rotations (45 points)
The rotational equations of motion for rigid body are
his (I2 13)w2w T1
(I1 12)wing T3,
where I1 12 2 la>0 represent the principal moments of inertia of rigid body about its
principal axes, W; represents the angular velocity about principal axis i, and T= represents
the moment exerted about principal axis
For the purposes of this question, we consider these equations of motion as system
à f(2) -1
with is and . = (wi.Wa,us)" Let (wi.0,0),19 (0,W2,0),23 (0,0,w3) be
three states corresponding to spins about each of the principal axes, respectively.
(a) (10 points) Show that, when =0. is an equilibrium of (4) for each i= 1,2,3
(b) (15 points) Linearize the equations of motion at each of the three equilibria from part
(a) to yield three linear systems à A.x. = 1,2,3. Which equilibria are stable?
Hint: 11 12 >I implies that 11 12 >0.
(c) (12 points) Keeping u 0. show that the origin is stable equilibrium of (4).
Is asymptotically stable?
(d) (8 points) Suppose that the moment inputs apply the feedback control Ti
where 81,82 hz are positive constants. This vields the losed loop system x= fcr(2
Show that the origin : =W O of the closed-loop system is an asymptotically stable
equilibrium for which the region of attraction is the entire state space. Hint: Use the
rotational kinetic energy T thwi I2W5 law3)
Comment: Theresult from part (d) part of the work one would do to design pointing
controller, for example to ma kes spacecraft point stably in particular orientation (Think
of space telescope pointing at distant galaxy in order to take an image.) Stabilizing
W =0 ensures that the spacecraft comes to rest. but does not guarantee anything about
the orientation the telescope will take rest. A full pointing (controller requires accounting
for the orientation in the control law.
Problem 2. State-space analysis of PD controller (55 points)
Consider the controlled inverted pendulum shown in Figure The pendulum can be
controlled by applying torque at the joint. The equation of motion and output equation
are given by
Figure 1: A controlled inverted pendulum.
(a) (8 points) Put the system equations in state space form with u =T and linearize
about 0=0=0 to vield an LTI system à A. Du.
(b) (10 points) Show that the equilibrium where o 0 Ois unstable when y = = 0 0.
(c) (10 points) Let u Kp for some constant gain K, > O. Write the closed-loop
system dynamics as , = Apr. For what values of Kp is the closed-loop system
stable? For what values of Kp is the closed-loop system asymptotically stable?
(d) (10 points) Add derivative term to your control law so that u = -Kp0 Kpb
for constant gains Kp, Kp > 0. Write the closed loop system dynamics with this
control law as 2 = Appz. For what values of KF and Kr is the closed-loo system
stable? What about asymptotically stable? Hint: &= -b+ Va has negative real part
if Va-0 <0.
(10 points) Let 9.81m/s² and pick values of Kp and Kp such that the
system Appa is asymptotically stable Solve the Lyapunov equation AT. ,P
PApr =-Q for some Q 0. Use the resulting positive-definite matrix Pto define
a function TPx. Show that v< Owhen . Appr.
(f) (7 points) Keep the PD control law. i.e. u -Kp8 KnP using the values of the
gains Kp. Kp from part (e). Suppose that the actuator is torque-limited such that
1-1 7. With this control law and actuator, the closed-loop linear system system
takes the form
where is the actual applied input from the actuator. defined as
Using the Lyapunov function V defined in part (e). compute V(s) for the system (5).
This can be done while leaving as a variable. but if you prefer you may take
For what states is V(r) <07 Use this result to sketch the set of states that can be
Comment. Parts (a)-(d) are very similar to the analysis of controller using classical
control theory techniques. e.g., computing poles of the system transfer function By solving
the Lyapunov equation in part (e) you found Lyapunov function which allowed you to
analyze the system with torque-limited actuator. This torque limited case is essentially
nonlinear and is very difficult to analyze using classical control theory techniques
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